Problem 61 - Calculus for Biology and Medicine

Question

Formal Definition of Limits: \lim _{n \rightarrow \infty} a_{n}=a .\( Find the limit \)a\(, and determine \)N\( so that \)\left|a_{n}-a\right|<\epsilon\( for all \)n>N\( for the given value of \)\epsilon$. $$ a_{n}=\frac{n}{n+1}, \epsilon=0.01 $$

Step-by-Step Solution

Answer

The limit is 1, and \( N = 99 \) is the value such that \( |a_n - a| < 0.01 \) for all \( n > N \).

Step 1: Find the Limit

To find the limit \( a \) as \( n \rightarrow \infty \) for \( a_n = \frac{n}{n+1} \), notice that as \( n \) becomes very large, the terms become dominated by \( n \). We simplify \( \frac{n}{n+1} \approx \frac{n}{n} = 1 \). Therefore, the limit \( a = 1 \).

Step 2: Set Up the Inequality

We need \( |a_n - a| < \epsilon \). Substitute \( a = 1 \) and \( a_n = \frac{n}{n+1} \) into the inequality: \[ \left| \frac{n}{n+1} - 1 \right| < 0.01 \] Simplify the expression: \[ \left| \frac{n - (n+1)}{n+1} \right| = \left| \frac{-1}{n+1} \right| \] which simplifies to:\[ \frac{1}{n+1} < 0.01 \]

Step 3: Solve the Inequality for N

The inequality \( \frac{1}{n+1} < 0.01 \) can be rewritten as:\[ n + 1 > 100 \] This simplifies to: \[ n > 99 \] Thus, \( N = 99 \). Therefore, for all \( n > 99 \), the inequality is satisfied.

Key Concepts

Formal Definition of Limits

The formal definition of limits provides a precise way to understand what happens to a sequence or function as it approaches a particular value. In essence, a limit describes the behavior of a function as its input approaches some value. For sequences, this typically means as the index tends to infinity. When dealing with sequences, the formal definition of a limit states that the sequence \(a_n\) converges to the limit \(a\) if, for every small positive number \(\epsilon\), there exists a corresponding integer \(N\) such that for all integers \(n > N\), the terms \(a_n\) are within \(\epsilon\) of \(a\).
In simple terms, as \(n\) becomes very large, \(a_n\) gets closer and closer to \(a\) and stays within \(\epsilon\) of \(a\). This is what it means for \(a_n\) to converge to \(a\). This concept is pivotal in calculus, helping us understand tendencies and the behavior of functions and sequences.

Epsilon-Delta Definition

The epsilon-delta definition of a limit is a formal method of proving that a specific value is indeed the limit of a function or sequence. This concept is vital for the precise formulation of limits in calculus. The definition utilizes two small quantities, \(\epsilon\) and \(\delta\), to establish that the function values can be made arbitrarily close to a specific value within a given ¬"distance".

• \(\epsilon\) represents the maximum error or approximation allowed.
• \(\delta\) is the distance from the limit value within which the function must remain.
  

In the context of sequences, \(\delta\) is replaced by an integer \(N\), which represents the point beyond which all terms of the sequence fall within the \(\epsilon\) distance of the limit \(a\).
Through steps like setting the inequality \(|a_n - a| < \epsilon\), we can determine an \(N\) such that for all \(n > N\), the inequality holds true. This process solidifies that \(a\) is the true limit of \(a_n\).

Inequalities in Calculus

Inequalities play a crucial role in calculus because they help us bound and estimate functions and sequences. They provide a fundamental tool to show convergence and analyze the behavior of sequences and their limits.

• An inequality like \(|a_n - a| < \epsilon\) is a mathematical way to express that \(a_n\) can be made arbitrarily close to \(a\).
• Using inequalities allows for establishing bounds, which are essential in proving the convergence of sequences.

In the context of this problem, inequalities help us identify a threshold value \(N\), ensuring that beyond \(N\), the terms of \(a_n\) lie within an \(\epsilon\)-neighborhood of \(a\). Mastering inequalities is essential as they are integral to solving limit problems and proving convergence in calculus.

Convergence Criteria

Convergence criteria are the conditions or rules that a sequence must satisfy to be considered as convergent. In simple terms, these criteria ensure that as we progress through the sequence, the terms get closer and closer to a specific value, known as the limit. There are several ways to test for convergence in sequences and series.
One vital criterion is the \(\epsilon\)-\(N\) condition used in the formal definition of limits. For the sequence \(a_n\) to converge to limit \(a\), for any small \(\epsilon > 0\), there must be an integer \(N\) such that for all \(n > N\), the inequality \(|a_n - a| < \epsilon\) holds.
This condition confirms that after a certain point in the sequence, every term is close to \(a\).
Understanding convergence criteria not only facilitates the study of sequences but also aids in grasping more complex concepts in calculus, such as infinite series and integral calculus. They form a cornerstone of calculus, ensuring that we can determine and trust the behavior of mathematical sequences and functions.